Problem 45 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ 1 \leq|x| \leq 4 $$

Step-by-Step Solution

Answer

The solution is \([-4, -1]\cup [1, 4]\).

Step 1: Understand Absolute Value

The absolute value \(|x|\) represents the distance of \(x\) from zero on the number line. Hence, \(|x| = x\) if \(x\geq0\), and \(|x| = -x\) if \(x<0\). We will use this property to solve the inequality \(1 \leq |x| \leq 4\).

Step 2: Split the Absolute Value Inequality

The inequality \(1 \leq |x| \leq 4\) implies two simultaneous possibilities: (1) \(1 \leq x \leq 4\) and (2) \(-4 \leq x \leq -1\). These inequalities represent two intervals on the number line.

Step 3: Solve the Positive Inequality

Consider the first inequality \(1 \leq x \leq 4\). This means \(x\) can be any number from 1 to 4, inclusive. Thus, the interval is \([1, 4]\).

Step 4: Solve the Negative Inequality

Now consider the second inequality \(-4 \leq x \leq -1\). This means \(x\) can be any number from -4 to -1, inclusive. Thus, the interval is \([-4, -1]\).

Step 5: Combine Intervals Using Union Notation

Since \(x\) can satisfy either of the two inequalities, we combine these two intervals using union notation. Therefore, the solution is the union of the two intervals: \([-4, -1] \cup [1, 4]\).

Key Concepts

Absolute Value

The concept of absolute value is key when dealing with inequalities involving distance on the number line. Absolute value, denoted as \(|x|\), is defined as the distance of a number from zero. This is always a non-negative value.

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

Understanding absolute value helps break down complex expressions into simpler parts. For example, with the inequality \(1 \leq |x| \leq 4\), we recognize that \(|x|\) can represent two conditions. These conditions help us describe the range of \(|x|\) in terms of ordinary values.

Interval Notation

Interval notation is a way of representing solutions to inequalities. It clearly outlines whether endpoints or boundaries are included in the solutions.

• Square brackets \[a, b\] indicate that the endpoints \(a\) and \(b\) are included in the interval, meaning they are part of the solution.
• Round brackets \((a, b)\) indicate that the endpoints are not included.

In the context of the inequality \(1 \leq |x| \leq 4\), the solutions are represented by the intervals \([1, 4]\) and \([-4, -1]\). This tells us that all numbers from 1 to 4, and from -4 to -1, are part of the solution.

Number Line

A number line is a visual tool used to represent numbers and understand inequalities better. It helps depict how easy it can be to see ranges and points marked with specific values.
Visualize the number line as a straight horizontal line.

• Highlighting the intervals \([-4, -1]\) and \([1, 4]\) helps us see which numbers satisfy the inequality conditions.
• Solid dots or continuous lines at 1, 4, -4, and -1 indicate inclusive ranges.

Using a number line ensures that we can see the separation of different intervals and better understand their real-world meaning.

Union of Intervals

When dealing with multiple intervals that solve an inequality, we often combine them with the union symbol (\( \cup \)). This notation shows all possible solutions that satisfy any part of the inequality.
For the inequality \(1 \leq |x| \leq 4\), the interval \([-4, -1] \cup [1, 4]\) represents all numbers in both ranges.

• The union includes every value from both intervals, showing that either range satisfies the original inequality.
• In practice, this means \(x\) can be any number from -4 to -1 or from 1 to 4.

Using union provides a concise, clear representation of all possible solutions in interval notation.